Nonadditive and additive ligand fields and spectrochemical series

Fluoro-diamine complexes of chromium(III). Joe W. Vaughn. Coordination Chemistry Reviews 1981 39 (3), 265-293. Kinetics and mechanism of the aquation ...
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Inorganic Chemistry, Vol. 15, No. 6, 1976 1399

Ligand Field Parameterization

Contribution from Chemistry Department I (Inorganic Chemistry), University of Copenhagen, H . C.&sted Institute, DK-2100 Copenhagen$, Denmark

Nonadditive and Additive Ligand Fields and Spectrochemical Series Arising from Ligand Field Parameterization Schemes. Pyridine as a Nonlinearly Ligating .rr-Back-Bonding Ligand toward Chromium(II1) J,dRGEN G L E R U P , OLE MjdNSTED, and C L A U S E R I K S C H A F F E R * Received July 17, 1975

AIC505040

In connection with the parameterization of the ligand field of a series of the trans-tetraammine- and truns-tetrakis(pyridine)chromium(III) complexes the nonadditive ligand field and the additive ligand field have been characterized. The operator of the former has the symmetry of the whole complex, while that of the latter consists of terms where each term has the symmetry of part of the complex. Within both the nonadditive and the additive ligand fields there are two types of parameterization, which are linearly related to each other. One has been called the crystal field parameterization; the other, the ligand field parameterization. The additive field of a linearly ligating ligand J is characterized by the parameters h ' , ~and h',J, which in the angular overlap model are interpreted as expressing the a-antibonding and n-antibonding (or bonding) effects upon the d orbitals caused by the interaction with the orbitals of J and consequently are renamed etoJ and e',J. Using the angular overlap model, as opposed to the electrostatic model, it is possible to treat nonlinearly ligating ligands such as pyridine in a quite analogous way to that used for systems of linearly ligating ligands. In this way the ligand field of pyridine has been handled. O n the basis of solution absorption spectra of the complexes using the angular overlap model and a transferability assumption for common single-ligand parameters, the two-dimensional spectrochemical series of ligands, corresponding to the above-mentioned ligand field parameterization, has been established for chromium(II1). T h e IT parameter associated with the chromium(II1) to pyridine bond has been found to be negative corresponding to net r-back-bonding, and the angle between the pyridine planes and the plane defined by the four nitrogen ligators has been found to be 3 8 O . A linear relationship between the enthalpies of activation for the hydrolysis of the ions [Cr(H,O),J]"+ and AJ has been established.

1. Introduction For years this laboratory has studied absorption spectra, circular dichroism spectra, and magnetic, kinetic, and thermodynamic properties of chromium(II1) complexes. The present paper is concerned with the ligand field spectra of a particular class of compounds called' orthoaxial complexes. We want to outline briefly the model approach. The word "field", contained in the concept of the ligand field model, would seem to imply that the effect of the ligands is represented by a field, i.e., that we have a perturbation model. We take this to mean a first-order perturbation model with a basis set consisting of five d functions. We refer to a previous paper2 about details and further references, recapitulating here only that a distinction between a nonadditive3 and an additive4 ligand field is useful. In the semiempirical nonadditive field the effect of all of the ligands in a complex is represented by a perturbation operator which has the symmetry properties of the complex but which cannot be divided into terms representing the individual ligands. The number of independent empirical parameters required to parameterize the field is determined exclusively by the symmetry of the system. In the additive field, on the other hand, the perturbation operator, or rather the nonspherical part of it, is written as a sum of terms representing the individual ligands. This implies the concept of ligand field parameters referring to the individual central ion to ligand bonds and opens up the possibility of transferability of such single-ligand parameters from one complex to another. It is one of the purposes of the present paper to discuss how such a transferability can be investigated experimentally. In order to compare the theory with experimental results for a dq system a model is required also for parameterizing the interelectronic repulsion. Here the usual first-order perturbation treatment is used with the same basis set, i.e., the usual Slater-Condon-Shortley treatment. The combination of the semiempirical ligand field model and the semiempirical repulsion model in this way has been called the expanded radial function model.5 In the present paper a

barycentered2 interelectronic repulsion parameter b will be used. Expressed in this parameter the energies h[4F] and h[4P] of the highest spin-multiplicity terms of the d3 configuration are -3b and 76, respectively. We then define the concept of orthoaxiality. This concept has been discussed previously in the additive ligand field when the orthoaxiality derives its name from the fact that the angles between neighboring central ion to ligand bonds are 90°. The characteristic property of the orthoaxial complexes is that the matrix elements between e,(&) and t2,(Oh) orbitals are vanishing and this property may be used as the basis for a redefinition of orthoaxiality in the nonadditive ligand field. The model basis for theoretical consideration of the spectra of such complexes was developed years ago in a series of papers which were hardly linked together by their own authors and which we therefore want to outline. It started with a thesis by Ilse,6 who used the additive fields, known as the point-charge and point-dipole models. This point-dipole model was later used to discuss explicitly orthoaxial c o m p l e x e ~ ,but ~ ~ ~the nonadditive field appeared shortly in a li and fieldg and in a crystal field'O,ll parameterization and the additive field appeared in a ligand field p a r a m e t e r i ~ a t i o n . ' ~ ~ ' ~ A relationship between the parameters of the nonadditive and the additive field was ivenI4 first in the crystal field parameterization and in the ligand field parameterization. Relationships between the two kinds of parameterization in the nonadditive field3,I7and in the additive also exist in the literature. However, the firm distinction between the nonadditive and the additive field is rather recent2 and so are pro osals for standardizing the choice of empirical parameters.33tls-20 As a consequence of the historical development indicated, a variety of parameters exists in the chemical and physical literature and it is often difficult to compare papers even on closely related subjects. It was soon realized that the energy of the lowest energy spin-allowed transition Azg(Oh) T2g(Oh) in octahedral d3 and ds complexes within the expanded radial function model

9

-

1400 Inorganic Chemistry, Vol. 15, No, 6, 1976

Schaffer et al.

was equal to the one-electron energy difference A between the spectra of crystals containing the same chromophoric come,(@) and t2g(Oh) orbitals. However, it was not until much plexes have often revealed shoulders as separate bands and later that it was found that the energies of the split components thereby confirmed their interpretation. Approximate positions of this transition due to an orthoaxial additive ligand field in of the individual components of overlapping absorption bands the pure cubic subconfiguration approximation depend on the have often simply been estimated from inspection of absorption A values for the individual ligands of the This curves. We believe that a curve analysis procedure is a more rule has been extremely important because it has made assatisfactory way of obtaining these positions and this is the signments of these split components completely unambiguous. reason for our choice. W e refer to the truly orthoaxial complexes that have apSo we consider our work, based upon a mass of undetailed peared in the l i t e r a t ~ r e ~ ~and - ~ Oto the studies concerning information and a generally applicable theoretical model, as almost orthoaxial c ~ m p l e x e s ~ of ~ - chromium(II1). ~* Since complementary to work concerned with detailed information 1968 it has fortunately become c ~ s t o m a r y ~to~use , ~ the ~ , ~ ~ and detailed calculations on individual systems. full energy matrices of the expanded radial function model B. The Nonadditive Field. From electron spin resonance in order to derive empirical parameters from experimental studies in this laboratory44 it could be concluded that all energy differences. trans-tetraammine- and -tetrakis(pyridine)chromium(III) In the study of chiral metal ion chromophore^^^ it was complexes, dissolved in glasses, have essentially tetragonal realized that they can be described as having a major orsymmetry. The zero-field splittings could largely be explained thoaxial ligand field, mainly but not solely responsible for the on the basis of the one-electron energy differences which in position of their energy levels, with a superimposed nonoragreement with the present work previously were found by us25 thoaxial field, responsible for the rotational strengths. We were on the basis of the absorption spectra of the trans-tetraammine interested in information about the major orthoaxial field and complexes. It was further shown in one instance by comparison with this purpose we p r e ~ a r e d members ~ ~ , ~ ~ of the achiral of the ESR spectra of [Cr(py)4Br2]+ in a glass and diluted trans-tetraamminechromium(II1) series to compare this with, in a crystal matrix of known structure45 that the magnetic tetragonal axis coincides with the molecular tetragonal axis, for example, the chiral trans-bis(trans-l(R),2(R)-cyclohexanediamine)chromium(III) series.42 as opposed to the case of a similar ethylenediamine complex.48 Later we became interested in the trans-tetrakis(pyriThis means that the holohedrized symmetry45is in all cases D4h so that the energy levels within the expanded radial dine)chromium(III) series43 which in addition to being orthoaxial has three properties of particular interest to ligand function model can be characterized by irreducible representations of this group. field studies. They are potentially chiral and the pyridine In general one may think about the ligand field operator molecule is a simple case of a nonlinearly ligating ligand16 V as consisting of an unobservable part of spherical3 symmetry (section 2C), which further has the possibility of n-backV(R3,), which has no influence upon energy differences within bonding. In the present paper the spectra of the tetraammine and the a dq configuration, and a remaining part 0. may be expressed as a sum of terms whose number is determined by tetrakis(pyridine) series have been analyzed simultaneously by introducing the transferability assumption together with ~ y m m e t r y . ~ W e shall speak about a ligand field parameterization3 when one uses the one-electron energies to the angular overlap model,I5 both of which associate with each define the parameters. This may be done either in terms of metal to ligand bond a c and a n contribution. Resolution a set of independent energy differences or in terms of the of the pyridine complexes into antipodes has not been acenergies measured relative to their average energy.3 Generally complished but the nonlinear ligation and the n-back-bonding one has in both cases to supplement the energy parameters toward the aromatic ligand pyridine have been illuminated and with parameters representing the symmetry-required nonvarious spectrochemical series have been established. diagonal elements of V. 2. The Ligand Field Such nondiagonal elements do not occur in orthoaxial A. The Chemical Approach to Spectroscopy. The usual complexes except when the lowest possible holohedrized experimental spectroscopic approach to transition metal symmetry D2h is attained. Even in this case it is possible to complexes seeks to accumulate as much information as possible define the e,(Oh) and t2,(Oh) orbitals out of the d set in a way about an individual chemical system. Similarly molecular which for practical purposes is unique. This is because the orbital calculations are always concerned with individual three perpendicular symmetry axes of D2h may also be taken systems. In the present paper the gain in generality which to define an octahedron. This is the basis for the strong-field arises from the restrictive assumptions of the expanded radial quantization used in the d 3 energy matrix (Table I) of the function model is used to treat a series of complexes simulexpanded radial function model. taneously. In tetragonal symmetry the two ligand field parameteriThe purpose of this approach is to try, particularly by zations may be expressed by the defining equations49 invoking the transferability assumption, to extract from a large amount of ligand field spectral material the chemically inh[e] (3/5)A(d) + (1/2)A(e) (14 teresting information it might contain. One might say it is h[B]E (3/5)A(d) - (1/2)A(e) tlb) a chemical approach to spectroscopy rather than a physical ZR1 L -(2/5>A(d) + (2/3>A(t2> (IC) one. We have been concerned with the visible absorption spectra h[q]=h[t;] E -(2/5)A(d) - (1/3)A(t,) (14 of solutions mainly of tetragonal chromium(II1) complexes where h[t], for example, is the diagonal element (tl0Ic)of the and have used a Gaussian analysis data reduction to represent ligand field operator. When this operator is written as a sum25 our experimental spectra. This is from a theoretical point of of a cubic term V(Oh) and two tetragonal terms V(D4he) and view a most doubtful procedure, and from a statistical point v(D4ht2),then the three terms are associated with the symof view it is rarely justified. However, a broad shoulder on metry parameters A(d), A(e), and A(t2), respectively. a broad absorption band of a chromium(II1) complex has Equation 1 contains the definitions of these parameters in always been taken as evidence for the existence of at least two terms of the one-electron energies. A(e) and A(t2) are seen electronic transitions and this interpretation has to our to be the tetragonal splitting parameters within the subshells knowledge always been borne out by the internal consistency eg and t2g and A(d) is the energy difference between the found for series of solution spectra. Furthermore, polarized

Inorganic Chemistry, Vof. 15, No. 6, 1976 1401

Ligand Field Parameterization

+

, +

+

-

3

F

A4

IZ

IZ +

x+

.E

average energies25 of these two subshells. The signs are of course important. It is further seen that the barycenter rule applies to the whole d shell. This is a way of expressing that the operator P contains no spherically symmetrical part. Analogously the validity of the barycenter rule fo! the e and t2* subshells txpresses that neither the operator V(D4hej nor the operator V(D4ht*)contains a cubically symmetrical part. Equation 1 may be inserted into the 10 X 10 energy matrix of Table I. Thereby this matrix is converted into a matrix which falls into blocks of 1 X 1 [Bl(D4h)], 1 X 1 [B2(D4h)l, 2 X 2 [A2(D4h)lr and 3 X 3 [ E ( D d l . It is readily verified that the barycenter rule for A(e) and A(t2) applies to the cubic parentage functions of T I and T type and for A(d) and b it applies to the whole d 3 or d configuration. The symmetry parameters A(d), A(e), and A(t2) of the ligand field parameterization are linearly related2 to any other internally linearly independent set of three symmetry parameters. The circumstance that a function set, in this case a d set, adapted to spherical symmetry is used as the zero-order functions suggegts an expansion of the perturbation Hamiltonian in terms whose electronic factors transform irreducibly under the three-dimensional rotation-inversion group R31. Such a procedure, which today rationally can be followed using 31 symbol^^^^^^^^ and the Wigner-Eckart theorem, has in fact been applied from the childhood of the crystal field model and we shall therefore call parameterizations of this type crystal field parameteri~ations.~We wish to stress that this naming does not imply that the parameters should be interpreted electrostatically. C, The Additive Ligand Field. When the ligand field operator is assumed to consist of a sum of terms, where each term represents the perturbation from one particular ligand we speak about the additive ligand field. In this case we write the operator as A and it has the form A = ;I: A(jJ)

8

4

*

5

8

-" , C. c

5 d

c

O

+

I

+

+

3

3 5 Q

BE

dJ)

E

& c O K

+ i E 82 +

+

+

+

L

B 3

+

HE

c

I*

+

B

B

I

5 P

2

+

-

I*

+

+

-

IF +

+

-

I=

+

where j within a chosen frame for the particular central ion to ligand system represents the position and rotational orientation of the ligand J so that UJ) is a set of variables which are bound together during the summation. The additive field has been discussed in earlier papers4>15$16 and will not be treated further here. However, one important property of A must be stressed. This operator can, like the nonadditive field operator V, be thought of as-consisting of a spherical part A(R3,) and a remaining part A. A(R3i) has no consequences for the application of the model and may therefore equally well be written V(R3j) to stress the fact that the assumption of additivity need not be made4 for the spherically symmetrical field term. In representing the additive field we shall use here the assumptions of the angular overlap model. This means that we use the single-ligand parameters Ag and AT to which we may add an extra subindex J to refer to the ligand J while we remember that in this work all parameters refer to chromium(II1). For linearly ligating ligands, Le., ligands for which the central ion to ligand system has the symmetry CmD, their formal expressions are4 J = 3(euJ - e8 J> (24 & J = ~ ( G J - ~ ~ ~ J ) (2b) where QJ (A = u, T , 6) are the one-electron energies of the ligand field model in the system consisting of the central ion M and a single ligand of type J. The expressions of eq 2 require only the assumptions of additivity and linear ligation in order to be valid within a d basis set. However, in the angular overlap model the parameters are interpreted as

1402 inorganic Chemistry, Vol. 15, No. 6, 1976

Schaffer et al.

associated with the involvement of the d electrons in the M-J bonding. So when 6 bonding is unimportant relative to a and T bonding, e,J - e6J is the a-antibonding energy of that d orbital which in the M-J system is du, Le., has rotational symmetry about the M-J bond. eTJ- e6J may be written as enJ - e6 J = (enJ

- e6 J)B + (enJ - e8 db

(3)

where the superindices a and b refer to antibonding and bonding. In eq 3 the a term is positive and the b term is negative, so that the net energetic consequence, which is the experimentally accessible quantity, may become negative if T back-bonding gives the dominating contribution to the dx-electron energy. Assuming the ligand ammonia to be effectively linearly ligating, the parameters of the nonadditive field for the system trans-[Cr(NH3)4AB]"+ may be expressed in terms of A, and A, parameters for the individual chromium(II1) to ligand bonds. Abbreviating A,NH~and AT^^, as Aux and an^, the parameters of the nonadditive field may be expressed in terms of those of the additive field as A(d) = '/6(&.4 - &A> + '/ 3 (AoN - AnN)

A(e> = -

' / 3 ( 4 ~

- &N>

A(tz> = - ' / 4 ( & ~ -

+ '/~(&JB- A ~ B ) (44

- '/3(&~

&N>-

'/4(&3

-A~N) - &N)

(4b) (4c)

It is seen that when N = A = B = J, then A(e) = A(t2) = 0 and A(d)

= A,,

--ATj= A J

which is the general expression for the cubic field parameter for the complex [MJ61n+ in the additive field model. It is also seen that the parameter A(d) for the cubic part of the nonadditive field in the additive field is equal to the average AJ for all the ligands52 of the complex [Cr(NH3)4AB]"+. Although the right-hand sides of eq 4 contain six parameters, only three linear combinations of these parameters may be determined from the experiment. If, however, the angular overlap model interpretation of parameters is valid, A=N should vanish. We therefore define new parameters measured relative to ATN AthJ= A ~ -J

an^

(A = U, IT)

We want to estimate the linear combinations defined through

A(d) = ' / ~ A ' U AB ' / ~ A ' ~ AfB2/3A'upy - 2/3A'my

(7a)

A(e) = - 2/3A'GAB + 2/jA'fJpY

(7b)

A(t2) =-'/~A'.AB i'/2Arnpy - '/Z(COS 2$)Anpy

(7 c>

corresponding to eq 6 for the [Cr(NH3)4AB]"+ system. To obtain these equations of which only the last equation differs from that of the ammonia case, eq 45 of ref 16 has been used. Again it must be emphasized that, except for the ATpy parameter, ATh has been used as the zero point for all the parameters of eq 7 . Therefore the situation is as follows. First, the symmetry parameters A(d), A(e), and A(t2) for each tetraammine or tetrakis(pyridine) complex can be determined from the experimental transition energies. Second, the three parameters A l U ~A',,AB, , and A t Tcan ~ ~ be calculated from eq 6 on the basis of A(d), A(e), and A(t2) for the [Cr(NH3)4ABIfl+ complex. Third, on the assumption that A',AB and A t n ~ ~ , determined from the [Cr(NH3)4ABIn+ spectra, take on the same values for [Cr(py)4ABIfl+, AtQ4, AtTpy,and ATpy(cos2$) can be calculated from eq 7 on the basis of A(d), A(e), and A(t2) for the [Cr(py)4ABIfl+complex (Table 11). [The angle can only be determined if an additional assumption is made about the value of A=N (Table IV).] Until now, all considerations have been concerned with [Cr(NH3)4AB]"+-[Cr(py)4AB]fl+ pairs, and the parameters A',AB and A l n averaging ~ ~ over A and B have been involved. An extension of the transferability assumption to bridge between different [Cr (NH3)4AB] [Cr(py)4AB] pairs using eq 5 and assuming Atuu,Alxpr, A'AA,and A'AB(A = u, T ) to be transferable leads to the data reduction (Table 111). Finally, the assumption of transferability of ATp,(cos 2$) gives the final data reduction (Table IV). D. Assignment of Absorption Bands. A few examples will show how certain assignments of the d-d transitions can be made on the basis of comparative spectral studies using chemical arguments about the variation in the angular overlap model parameters on going from one complex to another. The assignment of the split components of the first cubic parentage absorption band is a simple matter since their positions depend only on the A values1%22 of the individual ligands of the complex in question. Let us therefore focus attention upon the split components of the second cubic parentage band. The transitions are

+

"+-

4A2(0)4B,(D,)

and rewrite eq 4 as

+

"+

a4T,(0)a4A2(D,)

"A,(O),B, (D,)+ a4T,(O)b"E(D,)

A(d) = ' / 3 A l o ~-~' / ~ A ' ~ A-kB2 / 3 A r u ~

(64

A(e> = - ' / 3 A l a ~ ~ ' / 3 A r o ~

(6b)

(6c) We consider next the [Cr(py)4ABIfl+system of symmetry C4 and with A and B linearly ligating. The pyridine molecules form a four-bladed propeller around the A-Cr-B axis ( z axis) with the nitrogen ligators in the xy plane. The pyridine molecular plane forms an angle $ with the xy plane. Using the angular overlap model we assume that only two oneelectron single-ligand parameters are nonvanishing, eUpyand eTpy. The latter parameter is associated with the x orbitals whose node planes coincide with the pyridine molecular plane. W e then define A(t2) = - l / 2 n l n ~ ~

where the coefficient 2 arises from the above-mentioned restriction upon the 7 orbitals which participate in the metal-pyridine bonding, and express the symmetry parameters A(d), A(e), and A(t2) in terms of the additive field for [Cr(py)4ABI n+

Let us characterize them by their excited states and call them the A2 and E transitions. Let us consider first the series (Figure 1) of trans-tetraammine complexes of (OH)2, (HzO)(OH), and (H20)2. There is clearly a gradual decrease of the magnitude of the splitting as one goes from (OH)2 to (H20)2. Therefore if we look at the (OH)z and (OH)(H20) complexes alone, we can conclude in the first place that the A2 transitions correspond for both systems either to the high-energy component or to the low-energy component, since it is impossible, if an additive field is assumed, that A2 is the high-energy component in one of the complexes and the low-energy component in the other one as this would have required a larger splitting in the (H20)2 complex than in the (OH)(HzO) complex. Then it is only possible that A2 is the high-energy component in both the (OH)2 and the (OH)(HlO) systems since the opposite as-

Inorganic Chemistry, Vol. 15, No. 6, I976 1403

Ligand Field Parameterization

h; 4

~ ( h=)i= I:1Di exp[-((h - ~ , ) / C J ~ ) ~ ]

I

30'-

20 -

20 10 -

I

I

\

IH~OIIOH)

miz

I

I

,

1

I

inZmzI

I

.

I

,

1

I

I

I

.

Figure 1. Solution absorption spectra of the dihydroxo, the aquahydroxo, and the diaqua complexes of the tvans-tetraamminechromium(II1) series.

, n

20 -

10 -

300

350

400

1

I

I

450

500

550

1

.

600 nm

Figure 2. Solution absorption spectra of the dihydroxo, the fluorohydroxo, and the difluoro complexes of the trans-tetraamminechromium(II1) series.

signment of the split bands would have led to negative values for both A o 0 ~- A,,oH~ and A n 0 ~- A*oH,: a chemically unacceptable situation if the angular overlap model interpretation of the parameters shall be reasonable. Once the assignments have been made in this way for the (OH)2 complexes, it is obvious from Figure 2 that the A2-split components also have to be the high-energy ones for the ( 0 H ) F and F2 complexes. In a similar way one can, by comparative studies like the examples given, arrive at assignments in other cases. For d i f l ~ o r o and - ~ ~ dibromobis(ethy1enediamine)chromiU ~ ( I I I complexes )~~ unambiguous assignments have been made on the basis of polarized crystal spectra. So, having illustrated above how one can go about making assignments partially on the basis of chemical arguments, we are now able to go the other way around: from the spectroscopically assigned bands of the F2 complex (A2 at higher energy) we may now conclude from the additivity assumption applied to the spectra of Figure 2 that A2 also is the higher energy component of a4T1 for the (OH)(F) and (OH)2 complexes. In this way one obtains large values for the A t C oand ~ A'*oH parameters discussed above, and these values lend support to the angular overlap model interpretation of the empirical parameter^.^ 3. Method of Calculation All calculations of parameters dealt with in this paper were performed within the framework of regression analysis. This technique is excellently described in a number of textbooks and will consequently not be covered here. The calculations to obtain the final parameters from the initial visible absorption spectra were carried out in three distinct steps. In the first step the observed visible absorption spectra, measured at 5-nm intervals, were expressed as a sum of four h Gaussian curves,53 e.g.

For almost symmetrical spectral bands without indications of band splittings attempts at resolving these into more than one Gaussian component frequently resulted in difficulties as expected. Either the normal equations of the regression analysis procedure became singular or the results were such that one Gaussian component described the major part of the spectral band and the remaining Gaussian components described minor deviations from the true Gaussian shape of the spectral band. Such behavior is obviously unwanted and was overcome simply by adding extra information about acceptable Gaussian component heights to the normal equations. For an unresolved spectral band of height D, which is required by the model to represent two bands and thus to be expressed as a sum of two Gaussian curves, extra pseudoexperiments have been added. These state that the individual component heights have been measured with a mean value D l 2 and variance (D/20)2. This approach produced acceptable parameter vectors for all of the spectra treated in this paper, and as a test of the independence of this final parameter vector upon the extra pseudoexperiments the following conditions were used. Let a* be the minimum parameter vector calculated from all of the experiments and a the linear approximation to the minimum parameter vector calculated only from the genuine experiments. Let further A be the variance matrix calculated from a* but without the extra heig)t-eyecments; then sufficiently small values of P(x2< (a - a) A (a* a)) for rank A degrees of freedom were taken as an indication of independence between the calculated minimum parameter vector, a*, and the extra pseudoexperiments. For further calculations the desired parameter vector was assumed to be estimated with mean value a* and variance matrix A. The Gaussian type of data reduction is essential for the following calculations but unsatisfactory for at least two major reasons. First, none of the measured absorption spectra could be approximated to any degree of satisfaction with a function of this form when compared to the measuring accuracy. Second, even if this initial data reduction was acceptable from a statistical point of view, the following data reduction step, i.e., the exclusion of spectral band heights and spectral bandwidths from the total experimental material, necessitated by the unfortunate limitations of the ligand field model, places attempts upon a quantitative discussion of visible absorption spectra at a most undesirable level. With these obvious model deficiencies in mind, we have accepted the Gaussian shape data reduction and taken the Gaussian component positions to represent the vertical transition energies of the ligand field It was found convenient not to carry the Gaussian component positions as explicit parameters. Instead they were calculated from the three ligand field parameters A(d), A(e), and A(t2) and the single barycentered interelectronic repulsion parameter b by diagonalization of the appropriate energy matrix. Consequently the regression analysis procedure54 yielded directly the four above parameters55 together with Gaussian component heights and bandwidths. In the second step the three ligand field parameters were transformed into the parameters of the angular overlap model, either by eq 6 and I for the trans-tetraammine-transtetrakis(pyridine) pairs or by eq 6 alone for the transtetraamminebromochlorochromium( 111) ion as in this case the corresponding pyridine complex was not prepared. The results of these calculations are given in Table 11. In the final calculational step average single-ligand parameters were calculated from those of the individual species or pairs of species investigated. As a quantitative description of the measured absorption spectra was not possible with the

1404 Inorganic Chemistry, Vol. 15, No. 6, 1976

Schaffer et al.

Table 11. Angular Overlap Model Parameters (in kK) (Eq 6 and 7), Derived from the Solution Absorption Spectra of trans-Tetraammine and trans-Tetrakis(pyridine) Complexes' trans-[Cr(NH,), F, 1' trans- [Cr(py), F, 1' trans-[Cr(NH,),Cl, 1' trans- [ Cr(py), C1, ] trans-[Cr(NH,), Br, 1' trarrs-[Cr(py),Br,] trans-[Cr(NH,),FCl] * trans- [ Cr(py), FCl]

Anpy(c0S 2$)

A'UPY

A'npy

7.0

18.3

-2.0

-0.79

16.0

2.9

17.1

-2.9

-0.57

21.66

15.1

3.6

17.3

-2.8

-0.65

21.03

18.8

4.5

16.9

-3.4

-0.74

21.37

19.0

5.4

18.2

-2.2

-0.71

21.14

15.7

2.8

A'uN

A'uAB

20.84

22.3

21.10

A'~AB

+

+

trans-[Cr(NH, ), FBr]' trans-[Cr(py), FBr]' trans-[Cr(NH,), BrCl]

a See also discussions at the ends of sections 2C and 3.

Table 111. Spectrochemical Seriesa

J NH, A'uJ = AuJ - AnNH3 A'nJ = AnJ - A n N H 3 AJ

21.08 0 21.08

i:

0.07

f

0.07

PY

F-

c1-

Br'

17.4 i: 0.3 -2.3 i 0.3 19.71 i 0.15

22.2 f 0.2 6.82 i 0.11 15.39 i: 0.13

16.4 i 0.5 3.3 i: 0.6 13.15 f 0.10

14.4 i 0.3 2.0 i 0.4 12.42 i 0.16

The first two rows are the set consisting of the A u j and A n j series, also called the two-dimensional spectrochemical series. The third row is the usual spectrochemical series of AJ for cubic complexes which here has been derived solely o n the basis of tetragonal complexes. All parameters in the table are the same (within the standard deviations) as those found p ~ e v i o u s l yfor ~ ~the~ transtetraammine ~~~~ series except for A N H ~which was taken previously from the spectrum of the hexaamminechromium(II1) ion as 21.6 kK (cf. also discussions at the ends of sections 2C and 3). a

Gaussian component shape model, realistic error estimates of the parameters obtained by these computations were lacking. For further calculations it was however assumed that the correlation coefficients and the relative magnitudes of the estimated standard deviations were well defined. The standard deviations upon the mean values of the parameters in Tables I11 and IV are based upon this hypothesis and upon a common adjustment of the absolute size of all of the standard deviations upon the mean values of the parameters from the individual experiments to fulfill a x2 test a t the 50% confidence level for these latter data reductions. Computer programs were written in the Algol 6 dialect for the RC4000 computer, and the calculations were carried out a t the H. C.$rsted Institute. 4. Results and Discussion

In Table I1 are given individual values for angular overlap parameters for five pairs of tetraammine and tetrakis(pyridine) complexes (eq 6 and 7). The parameter A',y is through the model given directly as the energy of the transition 4B1, -, 4B2, ( D 4 h ) in all the tetraammine complexes, and the lack of constancy of this transition energy thereby becomes a direct measure of the deficiency of the additive ligand field. This deficiency can also be read out of Table I1 where Atgpyappears in the region between 16.9 and 18.3 kK and AITpy between -3.4 and -2.0 kK. The negative value of AITpy is of particular interest, because it corresponds within the conceptional framework of the angular overlap model (eq 3) to a net H back-bonding from chromium(II1) to the pyridine ligand. Even though the estimated values of this parameter are not in very good agreement, the parameter is consistently found to be negative. This is to our knowledge the first case of such a direct ligand field parametric piece of evidence for H back-bonding, although the small variation in A values for d6 hexacyanometalates [Mn(I), Fe(II), Co(III)] has been taken as evidence for x back-bonding to CN- a long time ago.56 In the first two rows of numbers of Table I11 the results shown in Table I1 have been collected using eq 5 to obtain average single-ligand parameters A',J and A',j. The value zero for A t T arises ~ from its definition. Variances upon and covariances between the data of Table 11, although not tabulated,

are of course essential for this calculation. From the estimated single-ligand parameters A ~ , Jand Ati,j, AJ = A',, - A t T j = A,J - A,J has been calculated and is given in the third row. The parameter values for F,C1-, and Br- are not different from our earlier one^,^^,^' but the results given here have been obtained in a much more objective way than previously and are now based also upon the spectra of the tetrakis(pyridine) series. A',N = AN was previously taken from the transition 4A2g 4T2s (Oh)of the hexaamminechromium(II1) ion but is in this work derived from the trans-tetraammine series only, It is, of course, a deficiency of the additive ligand field that the spectra of the tetraammines give a smaller average value for ANH, than does the spectrum of the hexaammine. In Table IV the information derived from the pyridine complexes through eq 7c has been included by making the additional geometrical assumption that a common value for $ may be used for all of the pyridine complexes. This assumption may be tested statistically and it was found to be justified. From the additional assumption of Ary = 0, $ is estimated to be 37.8 f 1.1', which is in fair agreement with the value of approximately 45' known from a crystal structure.46 Assuming ai,^ # 0 would be the same as allowing for a linear electrostatic field contributing to the perturbation from ammonia on top of the perturbation arising from the bonding. Such a discussion would require also an electrostatic contribution to the perturbation from pyridine and this has been excluded from the beginning by keeping only the H parameter associated with the H orbitals whose node plane coincides with the pyridine's molecular plane. It may be noted by inspection of eq 7 that pyridine is effectively linearly ligating when $ = 45'. W e wish to stress, however, that the mean value and standard deviation obtained for $ exclude this situation. We now return to the spectra of trans-[Cr(NH3)4AB]"+ and trans-[Cr(py)4AB]"+ complexes (A, B = Br, C1, F) on which the parameter values of Table IV are based. For all of these complexes we have calculated (Table V) the transition energies assooiated with the parameters of Table IV. The interelectronic repulsion parameter b was arbitrarily put equal to 1000 cm-I, but the partial derivatives of the transition

-

Inorganic Chemistry, Vol. 15, No. 6,1976 1405

Ligand Field Parameterization

Table lV.a Angular Overlap Model Parameters from Final Data Reduction (cf. Sections 2C and 3) 21.11 t 0.07 17.54 t 0.27 -2.00 f 0.20 22.16 f. 0.20 6.77 t 0.12 16.63 t 0.41 3.48 t 0.44 14.76 f. 0.27 2.50 2 0.39 -0.50 2 0.06

-0.244 -0.450 +0.890 -0.419 +0.702 +0.683 -0.410 +0.478 +OS74 -0.109 +OS38 +0.478 +0.018 +0.492 +0.411 -0.020 +OS59 +0.393 +0.132 +OS70 +0.301 c0.460 -0.196 -0.439

+0.757 +0.279 +0.069 +0.231 +0.152 +0.396 +0.405 +0.379 +OS28 -0.135 -0.152

+0.978 +0.425 +0.423 +0.069

+0.428 +0.443 +0.152

+0.955 +0.405

+0.528

This table is based upon the assumption that Anpy(cos 21) is common t o all complexes of the tetrakis(pyridine) series. The first column gives this parameter and the angular overlap model parameters, the second one gives their values (mean values t their standard deviations), and the rest of the table is the lower left part of the matrix of correlation coefficients. It is seen that only corresponding A L a jand A n j are strongly (and everywhere positively) correlated which means that their difference AJ = A,J - A n j is in most cases better determined than the individual parameters, as also indicated in Table 111. It is further seen that the angular overlap model parameter values, standard deviations taken into account, are the same in Tables 111 and IV. The value -0.50 for the parameter Anpy(cos 21) corresponds to I) = 37.8 t 1.1" when an^ is put equal t o zero. a

Table V. Transition Energies (kK) for trans-AB(NH,), and trans-AB(py), Complexesa AB

a4E 18.23 17.42 16.94 16.58 17.39 16.92 17.66 17.11 16.34 16.07 16.64

FF c 1 c1 Br F c1 F Br Br Br C1

,B2 21.11 19.05 21.11 19.05 21.11 19.05 21.11 19.05 21.11 19.05 21.11

b4E

a4A,

a [a4E]/ab

24.95 23.49 25.66 23.74 25.34 23.54 25.24 23.51 25.92 23.91 25.78

28.20 28.08 24.88 24.92 25.99 25.99 26.56 26.53 23.71 23.79 24.30

0.013 0.308 0.135 0.008 0.057 0.010 0.026 0.042 0.227 0.062 0.180

a[4B21/ab a [ b 4 E ] / a b

0 0 0 0 0 0 0 0 0 0 0

a[a4A,]lab

5.835 5.495 5.289 5.378 5.526 5.532 5.630 5.572 5.012 5.142 5.154

4.394 3.508 5.203 4.599 4.955 4.262 4.839 4.103 5.390 4.853 5.299

a Based upon the ligand field parameters of Table IV and using b = 1000 cm-'. In the first column the AB ligands are given. The results for the ammonia complexes are in each upper row with those for the pyridine complexes directly below. The transitions are characterized in the table by the irreducible representations associated with the excited levels in the group D,. The ground level is in all cases ,B, (D,).The partial derivatives of the transition energies with respect t o the repulsion parameters are also given.

energies with respect to b are also given in the table. Since t h e zero point for the repulsion energy is here the 4F multiplet term and since the 4P term has the energy lob, a derivative with respect to b of a given level, multiplied by 10, is a measure of the ercentage of 4P character in that level. The value 1000 cm- for b is rather close to the values found for all complexes studied. It should be mentioned that this parameter is rather poorly determined in the present data material and therefore no attempt could be made at using this material to reestablish the nephelauxetic series.j6 On comparison of the energies of corresponding transitions in tetraammine and tetrakis(pyridine) complexes it is apparent that those whose excited levels are a4E and a4A2 have much the same energies in the two series although certain irregularities exist for a4E. On the other hand the transitions whose excited levels are 4B2 and b4E have quite different energies but nearly the same energy difference of about 2 kK both for 4B2 and b4E. The ligands water and hydroxide were included in our previous ~ o r k . * j , ~However, ' they are not linearly ligating and therefore there is not enough geometrical knowledge available to treat their complexes properly by the angular overlap model. Further their ESR spectra44indicate rhombic distortions. We tried to handle the aqua and hydroxo complexes under the assumption that they behaved effectively as linearly ligating and succeeded in part with the aqua complexes. The values of the parameters for water were A',,H~o = 23.7 f 1.8 and A ' . r r ~=z 7.7 ~ f 2.0 giving A H ~ O= 16.0 f 0.3 which is 1.4 kK smaller than the value observed for the hexaaquachromium(II1) ion. The very high value of A:H~o, m% indicate that the oxygen is trigonally or planarly ligating, Le., coordinating in such a way that the chromium(II1) ion lies in the plane of the water molecule. We have also analyzed

P

the ~ p e c t r a j ~of- ~all~ the ten isomers of the ammineaquachromium(II1) series, again by assuming both ammonia and water to be linearly ligating. In this case we found A',,N = 21.8 f 0.1, A',,H~o= 20.3 f 0.9, AlT~,0= 3.0 f 0.9, and A H ~ O = 17.3 f 0.1. Here the three complexes [ C ~ ( H 2 0 ) 6 ] ~ + [Cr(NH3)6] 3+, and fac- [Cr(NH3) 3( H20) 31 3+ (holohedrized symmetry o h ) , of course, contribute strongly to A H ~ Oand A N H ~ .The much lower value of A a ~ * found 0 here may indicate that the oxygen in these complexes is partially tetrahedrally ligating.16 This apparently strange behavior may tentatively be associated with differences in cation-solvent or cation-anion interactions since in the tetraammine and tetrakis(pyridine) cases where A ' T ~=27.7 ~ kK the major amount of experimental material is based upon dipositive chromium(II1) complexes, whereas in the case of the aquaammine series ( A ' T ~=z 3.0) ~ only tripositive chromium(II1) complexes contribute. In this connection it should be emphasized that all of the ions of Tables 11-IV have unit positive charge. The value A N H ~= 21.1 found for these 1+ ions, compared with ANH, = 21.8 found for the 3+ ions, may similarly be caused by different interactions with the second sphere. That fact that the single-ligand parameters for F, C1-, and Br- were found to be the same in this work as in our previous ~ o r k , even ~ ~though , ~ ~ the hexaammine had previously been used to fix A',,NH,, is understandable from our present calculations by the fact that this parameter was found to be little correlated with the other single-ligand parameters (Table IV). The analysis of the hydroxo complexes by the Gaussian procedure failed to such an extent that it would be unreasonable to give parameter values on this basis. When we did obtain parameter values in our previous work, it was because we fixed the energy of the transition 4Blg 4B2g (D4h). This transition never appears separated in the hydroxo complexes but according to the additive field model its energy should

-

Schaffer et al.

1406 Inorganic Chemistry, Vol. 15, No. 6 , 1976 Table VI. Additive Ligand Field Parameters for Linearly Ligating LigandsTOrthoaxial Complexes

Perumareddi parameters Yamatera parameters

McClure parameters

6,

AN

6,

4Sn

(5/2)~ ~ D ~ N . AD B t ~

,~ D~S N~. A R 1 0 0 0 1 0 0 1 0 0 1 1 0 1 3 1 1 1 0 -4 1 -1 0 0 1 0 0 a The parameters of the present paper (eq 5 ) are expressed in terms of parameter sets frequently used in the literature. Each parameter of the set {Dq, Ds, D t } has been supplemented with the subindex N,AB to indicate that it is used as an additive field parameter. With the which B using the table gives AAB = A t o -~A ~' n =~1 ~0 D q ~ ,+ ( 3 5 / 2 ) D t ~ ~ ~ ~ transferability assumption one then has D q N , = D ~ N ~ A which is a slight generalization of the expression of Wentworth and Piper." AN

46 0

A l o = ~ AN A'~AB A'TAB

equal A I u ~ .However, in the Gaussian analysis description this transition tends to move several kK toward the red. The spectra of the hydroxo complexes are not easy to understand even if nonlinear ligation of OH- is taken into account. The split components of the second spin-allowed band 4A2g a4T1, (Oh)often appear well separated and not excessively broad [Figure 11 while the first spin-allowed band 4A2g 4T2, (Oh) a t the same time is poorly split and cannot be represented by two Gaussians. If this behavior of the first band were caused by a superposition of spectra for molecules with hydroxide ions in different rotational orientations with respect to the chromium(II1) to oxygen bond, one would have expected similar split behaviors60 of the two first cubic spin-allowed transitions 4A2g 4T2g(Oh)and 4A2s a4T1, (Oh).In spite of these problems we still believe that our previous v a l ~ e s ~ .AI~oOH ~,~' = 26 kK and A',oH = 9 kK, although not valuable in a quantitative sense, are meaningful in attributing to the ligand OH- the highest values for both parameters. The well-known AJ spectrochemical series previously based upon spectra of octahedral complexes has here been reestablished on the basis of tetragonal complexes. In addition to this series we have obtained the A',J and A',J series which have previously been called the two-dimensional spectrochemical series,13 Le.

-

-

--+

+

< C1- < F - < H,O < py < NH, Br- < C1- < py < H,O(?) < NH, < F - < OH-(?) py < NH, < Br- < C1- < H,O(?) < F - < OH-(?)

AJ: BrA',J: A',J:

It must be emphasized that even with the bonding interpretation of the parameters of the angular overlap model the parameters refer only to the nonspherical part of the Hamiltonian and only to the bonding which is associated with the d orbitals, which means that even if the parameters have a chemical significance, they may not have a chemical importance. Therefore it is interesting to note that the enthalpies of activation for the aquation processes59,61,63-65 [Cr(OH,),J](3-i)'

+ OH,

+

[Cr(OH,)6]3++ Ji-

accurately, except for fluoride for which special circumstances operate,61s62follow the AJ series as depicted in Figure 3. This correlation and similar correlations for other systemss6 do point toward a chemical importance of such parameters, and this will be the subject of a more detailed report later. Appendix

c T

28

zsc

I

/

12

1 16

20

A I kcm-' 1

Figure 3. Empirical linear correlation between the enthalpies of activation AH* fo; the processes [CI(OH,),J](~-')' + OH 2 [Cr(OH,),l3' + J z - and AJ. Error bars are drawn as r the estimated standard deviations.

used ligand field parameterization schemes, only discussed shifts, and their parameters have therefore been supplemented with the parameter AN which expresses the energy difference h [ ~-] h [ ( ] in the parent chromophore [MNsl. The parameter Dq out of the set67,6*{Dq,Ds, Dt) also refers to the [MNs] chromophore and the relationship AN = 1oDqMx, is valid. The consequence is that Dq, even though its corresponding operator2s3 has cubic symmetry, does not refer to the average cubic field in the chromophore [MN4AB] but incorporates a field component of cylindrical symmetry. Similarly the supposedly cylindrical field parameter Dt contains cubic field components. These unfortunate circumstances result in the breaking down of the barycenter rule for Dt contributions when the energies of the tetragonal-split components of e,(&) and t2g(Oh) are expressed in terms of the set (Dq, Ds, Dt). Actually this set belongs to the nonadditive field. It corresponds to three linearly independent operators and is as such complete even though this has been refuted.20 The unfortunate property of the set of operators is that those corresponding to Dq and Dt are non0rthogonal.6~ Registry No.

trans-[Cr(NH3)4Fl]+, 31253-66-4; trans-[Cr-

(py)4F21f, 47514-84-1; trans-[Cr(NH3)4C12]+, 22452-49-9; Usually complexes parameterized as orthoaxial ones are trans-[Cr(py)4C12]+, 5 1266-53-6; tram-[Cr(NH3)4Br2]+, 51266-63-8; handled using the additive ligand field. In Table VI the additive field parameter set of eq 5 { A I u ~A, l U ~ A~ I, T ~ ~ ] , trans-[Cr(pya Br#, 5 1266-52-5; trans-[Cr(NH3)4FCI]+, 44437-04-9; rram-[Cr(py)4FCl]+, 5 1266-55-8; trans-[Cr(NH3)4FBr]+, 51 266-64-9; referring to the chromophore [MN4AB], is expressed in terms t r n n s - [ C r ( p y ) 4 F B r I f , 51266-54-7; trans-[Cr(NH3)4BrCI]', of the parameters most frequently used in the current liter22452-50-2; trans-[Cr(NH3)4(OH),]+, 51266-65-0; trans-[Crature. Common to all of the parameterization schemes is the (NH3)4(H20)(OH)] 3 1564-04-2: trans- [ Cr(NH3)4(H20)2] '+, fact that they are based upon shifts in the d-electron energies 36834-73-8; trans-[Cr(NH3)4F(OH)]+. 588 16-90-3; [Cr(OH2)5BrI2+, on going from the parent chromophore [MNs] to the actual 26025-60-5; [Cr(OH2)5Cl]*', 14404-08- 1 ; [ C ~ ( O H ~ ) S*+, F] 19559-07-0; [Cr(OH2)6I3+, 14873-01-9; [ C r ( 0 H 2 ) 5 ( p y ) l 3 + , chromophore [MN4AB]. Yamatera12 and McClure,13 who

'+,

Inorganic Chemistry, Vola15, No. 6,1976 1407

Ligand Field Parameterization 34714-27-7; [Cr(0H2),(NH,)l3+,

42402-03-9.

References and Notes ( I ) C. E. Schaffer and C. K. Jorgensen, K . Dan. Vidensk.Selsk., Mat.-Fys. Medd., 34, No. 13 (1965). (2) C. E. Schaffer in “Wave Mechanics-The First Fifty Years”, W. C. Price, S. S. Chissick, and T. Ravensdale, Ed., Butterworths, London, 1973, Chapter 12. (3) C. E. Schaffer, Theor. Chim. Acta, 34, 237 (1974). (4) C. E. Schaffer, Srruct. Bonding (Berlin), 14, 69 (1973). (5) C. K. Jorgensen, Discuss. Faraday Soc., 26, 110 (1958). (6) F. E. Ilse, Dissertation. Frankfurt am Main, 1946. (7) C. J. Ballhausen and C. K. Jorgensen, K . Dan. Vidensk. Selsk., Mat.-Fys Medd., 29, No. 14 (1955). (8) C. J. Ballhausen, F. Basolo, and J. Bjerrum, Acta Chem. Scand., 9,810 (1955). (9) J. S. Griffith and L. E. Orgel, J . Chem. SOC.,4981 (1956). ( I O ) W. Moffitt and C. J. Ballhausen, Annu. Reu. Phys. Chem., 7, 107 (1956). (1 1) C. J. Ballhausen and W. Moffitt, J . Inorg. Nucl. Chem., 3, 178 (1956). (12) H. Yamatera, Bull. Chem. SOC.Jpn., 31, 95 (1958). (1 3) D. S. McClure, “Advances in the Chemistry of Coordination Compounds”, S. Kirschner, Ed., Macmillan, New York, N.Y., 1961, p 498. (14) T. S. Piper and R. L. Carlin, J. Chem. Phys., 33, 1208 (1960). (15) C. E. Schaffer and C. K. Jorgensen, Mol. Phys., 9, 401 (1965). (16) C. E. Schaffer, Sfruct. Bonding (Berlin), 5, 68 (1968). (17) J. S. Griffith, Mol. Phys., 6, 503 (1963). (18) S. E. Harnung and C. E. Schaffer, Struct. Bonding (Berlin), 12, 257 (1972). (19) J. C. Hempel, J. C. Donini, B. R. Hollebone, and A. B. P. Lever, J. Am. Chem. SOC.,96, 1693 (1974). (20) J. C. Donini, B. R. Hollebone, G. London, A. B. P. Lever, and J. C. Hempel, Inorg. Chem., 14, 455 (1975). (21) Y. Tanabe and S. Sugano, J . Phys. SOC.Jpn., 9, 753 (1954). (22) R. A. D. Wentworth and T. S. Piper, Inorg. Chem., 4, 709 (1965). (23) D. M. L. Goodgame, M. Goodgame, M. A. Hitchman, and M. J. Weeks, Inorg. Chem., 5, 635 (1966). (24) D. A. Rowley and R. S. Drago, Inorg. Chem., 6, 1092 (1967). (25) J. Glerup and C. E. Schaffer, “Progress in Coordination Chemistry”, M. Cais, Ed., Elsevier, Amsterdam, 1968, p 500. (26) D. A. Rowley and R. S. Drago, Inorg. Chem., 7, 795 (1968). (27) C. W. Reimann, J . Phys. Chem., 74, 561 (1970). (28) A. B. P. Lever and B. R. Hollebone, Inorg. Chem., 11, 2183 (1972). (29) M. A. Hitchman, Inorg. Chem., 11, 2387 (1972). (30) A. F. Schreiner and D. J. Hamm, Inorg. Chem., 12, 2037 (1973). (31) W. A. Baker and M. G. Phillips, Inorg. Chem., 5, 1042 (1966). (32) L. Dubicki and R. L. Martin, Aust. J . Chem., 22, 839 (1969). (33) L. Dubicki, M. A. Hitchman, and P. Day, Inorg. Chem., 9, 188 (1970). (34) M. Keeton, B. Fa-Chun Chou, and A. B. P. Lever, Can. J . Chem., 49, 192 (1971); erratum, ibid., 51, 3690 (1973). (35) L. Dubick!,and P. Day, Inorg. Chem., 10,2043 (1971). (36) H.-L. Schlafer, M. Martin, and H.-H. Schmidtke, Ber. Bunsenges. Phys. Chem., 75, 787 (1971). (37) D. A. Rowley, Inorg. Chem., 10, 397 (1971). (38) T. J. Barton and R. C. Slade, J. Chem. SOC.,Dalron Trans., 650 (1975). (39) C. E. Schaffer, Proc. R.,Soc. London, Ser. A, 297, 96 (1967). (40) J . Glerup and C. E. Schaffer, Chem. Commun., 38 (1968). (41) J. Glerup and C. E. Schaffer, Inorg. Chem., following paper in this issue. (42) C. E. Schaffer and J. Glerup, Proc. Int. Conf: Coord. Chem., 9, 113 (1966). (43) J. Glerup and C. E. Schaffer, to be submitted for publication in Inorg. Chem. (44) E. Pedersen and H. Toftlund, Inor Chem., 13, 1603 (1974). (45) From a crystal structure analysiJ6 on trans-dibromotetrakis(pyridine)rhodium(III) bromide hexahydrate, which is isomorphic with the corresponding chromium(lI1) salt, it is known that the four pyridine molecules form a propeller around the Br-Rh-Br axis with a dihedral angle of approximately 45O between the plane formed by the four nitrogen donors and each of the pyridine molecular planes. So the s mmetry of [Cr(py)4A2] can be assumed to be 0 4 , and the ho1ohedrizedl.d; symmetry D4h both in the nonadditive and the additive field. On the other hand, the symmetry of Cr(py)4AB is only C4 and the holohedrized symmetry in the nonadditive field accordingly C4 X S2 = C4h. If, however, a partial additivity is assumed and the field is considered as consisting of the sum of that of the tetragonal propeller system Cr(py)4 of symmetry D4 with

(46) (47) (48) (49)

(50)

(51) (52) (53) (54)

(55)

(56) (57) (58) (59) (60)

(61) (62) (63) (64) (65) (66) (67) (68)

(69)

holohedrized symmetry D4h and the linear system AB of symmetry C,, with holohedrized symmetry D.h and the m axis coinciding with the fourfold axis of Cr(py)4, then the holohedrized symmetry is Dqh also here. The difference between D4k and C4h is for our purpose that blg(&,) and bzg(D4h) both transform as bg(C4h), but it is hard to see how the interaction between Cr(py)4 and AB should cause the one-electron functions of blg(D4h) and bZg(D4h) symmetry to mix. We therefore take our effective symmetry as D4h for all of the trans complexes investigated here. S. Yde-Andersen, K. J. Watson, and E. Pedersen, to be submitted for publication. C. E. Schaffer, Pure Appl. Chem., 24, 361 (1970). B. R. McGarve , J Chem. Phys., 41, 3743 (1964). 0 and t usuallySBmean ‘ a set of functions, which need not be d functions but which span e(@, is., the irreducible representation of the octahedral rotation group 0, by identically the same matrices as do drz and d,2-,2. Similar!y 1; v, and.5 span tz(0) as do d, d,,, and dyr, respectively. The d functions are said5 to be standard basis functions for the irreducible representations e(0) and tZ(0) and similarly the set d,,, d,, is a standard set for E)(Dm)and for the first E representation of all the groups D,. It is useful to distinguishsi between the restricted concepts of standard functions (which define Clebsch-Gordon coefficients and 3 - r functions with sign) and the more general concept of functions transforming the standard way. The set -p , p,, for example, transforms the standard way as e(&) whiled,, andvd,, are the standard functions which define the 3-r symbols.s1 J. S. Griffith, “The Theory of the Transition Metal Ions”, Cambridge University Press, London, 196.1. S. E. Harnung and C. E. Schaffer, Struct. Bonding (Berlin), 12, 201 (1972). 0. Bostrup and C. y. Jorgensen, Acta Chem. Scand., 11, 1223 (1957). C. E. Schaffer, Proc. Int. Conf. Coord. Chem., 8, 77 (1964). Since numerical differentiation of eigenvalues is sometimes encountered in the chemical literature, it is perhaps, in connection with the regression analysis procedure, worth it to mention explicitly that if a matrix, M, is diagonalized by QTMQ = L, then partial derivatives of the eigenvalues with respect to some variable, CY,say, are simply the diagonal elements of the matrix QT(d,M)Q where d, is an operator that replaces each element in M by its partial derivative with respect to the variable CY. In our earlier the parameters were obtained in a different way. From the four transition energies, which were obtained either just by inspection of the absorption curves or by Gaussian analysis, the parameters were obtained by the solution of a set of four nonlinear equations. This was accomplished by Newton’s method. The present approach has, besides the rigorous fashion in which the transition energies are obtained, the added advantage that the number of spectral transitions may be greater than the number of parameters without altering the scheme of calculation. C. E. Schaffer and C. K. Jorgensen, J. Inorg. Nucl. Chem., 8, 143 (1958). L. Monsted and 0. Monsted, Acta Chem. Scand., 27, 2121 (1973). L. Monsted and 0.Monsted, Acta Chem. Scand., Ser. A , 28,23 (1974). L. Monsted and 0.Monsted, Acta Chem. Scand., Ser. A, 28,569 (1974). This expectation is based upon the fact that the transition 4A2,(@J4B (D4h) 4T2g(Ok)4Eg(D4k)is associated with d,, -(1/2)d,y + (31/1$2)d,z and dyz -(1/2)d,2- 2 - (31/2/2)d,2 and the transition 4A2g(Oh)4B (D4h) a . 4 ~ (@J4E,(D4h) ~ is associated essentially with -(31~/2)dx2-y2- (lf2)d,2 and d,, +(31/2/2)dx2-y2- (1/2)d,2 d,, so that the orbitals d,, and d,, which are the only orbitals whose energies are sensitive toward the rotational orientation of the hydroxide ion, contribute in much the same way to the energies of the transitions to the two 4Eg(D4k)levels whose octahedral parentages are 4T2g(Oh)and a4Tl (Oh). T. Swaddle and E. L. King, Inorg. Chem., 4, 532 (1965). T. W. Swaddle, Coord. Chem. Rev., 14, 217 (1974). A. Bakac and M. Orhanovic, Inorg. Chem., 10, 2443 (1971). D. R. Stranks and T. W. Swaddle, J.Am. Chem. Soc., 93,2783 (1971). F. A. Guthrie and E. L. King, Inorg. Chem., 3, 916 (1964). S. T. D. Lo and T. W. Swaddle, Inorg. Chem., 14 1878 (1975). J. R. Perumareddi, J . Phys. Chem., 71, 3144 (1967). We use the sign convention of P e r ~ m a r e d deven i ~ ~ though the parameter set was originally introduced by Moffitt and Ballhausen.io$’l It should be noted that Perumareddi changed the signs on Ds and Dr but not on Dq. C. E. Soliverez, Int. J . Quantum Chem., 7, 1139 (1973).

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